The axiom of choice

Content created by Fredrik Bakke, Egbert Rijke, Jonathan Prieto-Cubides and Louis Wasserman.

Created on 2022-02-16.
Last modified on 2025-08-14.

module foundation.axiom-of-choice where

Imports

open import elementary-number-theory.natural-numbers

open import foundation.action-on-identifications-functions
open import foundation.coproduct-types
open import foundation.dependent-pair-types
open import foundation.function-extensionality
open import foundation.functoriality-propositional-truncation
open import foundation.inhabited-types
open import foundation.postcomposition-functions
open import foundation.projective-types
open import foundation.propositional-truncations
open import foundation.sections
open import foundation.split-surjective-maps
open import foundation.surjective-maps
open import foundation.unit-type
open import foundation.univalence
open import foundation.universe-levels

open import foundation-core.equivalences
open import foundation-core.fibers-of-maps
open import foundation-core.function-types
open import foundation-core.functoriality-dependent-pair-types
open import foundation-core.identity-types
open import foundation-core.precomposition-functions
open import foundation-core.sets

open import univalent-combinatorics.counting
open import univalent-combinatorics.finite-types
open import univalent-combinatorics.standard-finite-types

Idea

The axiom of choice^¶ asserts that for every family of inhabited types B indexed by a set A, the type of sections of that family (x : A) → B x is inhabited.

Definition

Instances of choice

instance-choice : {l1 l2 : Level} (A : UU l1) → (A → UU l2) → UU (l1 ⊔ l2)
instance-choice A B =
  ((x : A) → is-inhabited (B x)) → is-inhabited ((x : A) → B x)

Note that the above statement, were it true for all indexing types A, is inconsistent with univalence. For a concrete counterexample see Lemma 3.8.5 in [UF13].

The axiom of choice restricted to sets

instance-choice-Set :
  {l1 l2 : Level} (A : Set l1) → (type-Set A → Set l2) → UU (l1 ⊔ l2)
instance-choice-Set A B = instance-choice (type-Set A) (type-Set ∘ B)

level-AC-Set : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2)
level-AC-Set l1 l2 =
  (A : Set l1) (B : type-Set A → Set l2) → instance-choice-Set A B

AC-Set : UUω
AC-Set = {l1 l2 : Level} → level-AC-Set l1 l2

The axiom of choice

instance-choice₀ :
  {l1 l2 : Level} (A : Set l1) → (type-Set A → UU l2) → UU (l1 ⊔ l2)
instance-choice₀ A = instance-choice (type-Set A)

level-AC0 : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2)
level-AC0 l1 l2 =
  (A : Set l1) (B : type-Set A → UU l2) → instance-choice₀ A B

AC0 : UUω
AC0 = {l1 l2 : Level} → level-AC0 l1 l2

Properties

Every type is set-projective if and only if the axiom of choice holds

is-set-projective-AC0 :
  {l1 l2 l3 : Level} → level-AC0 l2 (l1 ⊔ l2) →
  (X : UU l3) → is-set-projective-Level l1 l2 X
is-set-projective-AC0 ac X A B f h =
  map-trunc-Prop
    ( ( map-Σ
        ( λ g → ((map-surjection f) ∘ g) ＝ h)
        ( precomp h A)
        ( λ s H → htpy-postcomp X H h)) ∘
      ( section-is-split-surjective (map-surjection f)))
    ( ac B (fiber (map-surjection f)) (is-surjective-map-surjection f))

AC0-is-set-projective :
  {l1 l2 : Level} →
  ({l : Level} (X : UU l) → is-set-projective-Level (l1 ⊔ l2) l1 X) →
  level-AC0 l1 l2
AC0-is-set-projective H A B K =
  map-trunc-Prop
    ( map-equiv (equiv-Π-section-pr1 {B = B}) ∘ tot (λ g → htpy-eq))
    ( H ( type-Set A)
        ( Σ (type-Set A) B)
        ( A)
        ( pr1 , (λ a → map-trunc-Prop (map-inv-fiber-pr1 B a) (K a)))
        ( id))

The axiom of choice fails to hold for arbitrary types. Indeed, it already fails to hold for the 0-connected 1-type $B Z_{2}$ as demonstrated in Corollary 17.5.3 of [Rij22]. This counterexample is formalized in foundation.global-choice. Hence it is both incompatible with univalence and with the existence of higher inductive types to assume the axiom of choice for all types.

References

[Rij22]: Egbert Rijke. Introduction to Homotopy Type Theory. December 2022. arXiv:2212.11082.
[UF13]: The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Institute for Advanced Study, 2013. URL: https://homotopytypetheory.org/book/, arXiv:1308.0729.